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Revisiting Bresse-Timoshenko theory for beams
Title
Revisiting Bresse-Timoshenko theory for beams.
Creator
Hache, Florian, Elishakoff, Isaac, Challamel, Noël, Graduate College
Abstract/Description
In this study, a variational derivation of the simpler and more consistent version of Bresse-Timoshenko beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of the different energy contributions is established, including the shear effect and the...
Show moreIn this study, a variational derivation of the simpler and more consistent version of Bresse-Timoshenko beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of the different energy contributions is established, including the shear effect and the rotary inertia. Second, the Hamilton’s principle is applied demanding the stationary of an appropriate functional, leading to two different equations of motion. The resolution of these equations provides the governing differential equation. It turns out that an additional term appears. The derived equations are intended for dynamic stability applications. Specifically, the parametric vibrations will be studied when the axial force varies periodically. This problem has important aerospace applications.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00005880
Format
Document (PDF)
Vibration of nonlocal carbon nanotubes and graphene nanoplates
Title
Vibration of nonlocal carbon nanotubes and graphene nanoplates.
Creator
Hache, Florian, Elishakoff, Isaac, Challamel, Noël, Florida Atlantic University, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
Abstract/Description
This thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional Bresse-Timoshenko models for thick beams and Uflyand-Mindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first...
Show moreThis thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional Bresse-Timoshenko models for thick beams and Uflyand-Mindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first one is derived through the use of equilibrium equations and leads to a truncated governing differential equation in displacement. It is shown, by considering a power series expansion of the displacement, that this is asymptotically consistent at the second order. The second theory is based on slope inertia and results in the truncated equation with an additional sixth order derivative term. Then, these theories will be extended in order to take into account some scale effects such as interatomic interactions that cannot be neglected for nanomaterials. Thus, different approaches will be considered: phenomenological, asymptotic and continualized. The basic principle of continualized models is to build continuous equations starting from discrete equations and by using Taylor series expansions or Padé approximants. For each of the different models derived in this study, the natural frequencies will be determined, analytically when the closed-form solution is available, numerically when the solution is given through a characteristic equation. The objective of this work is to compare the models and to establish the eventual superiority of a model on others.
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Date Issued
2018
PURL
http://purl.flvc.org/fau/fd/FA00013019
Subject Headings
Carbon nanotubes, Graphene, Vibration
Format
Document (PDF)
Some topics related to the vibrations of deterministic and non-deterministic metamaterial structures
Title
Some topics related to the vibrations of deterministic and non-deterministic metamaterial structures.
Creator
Li, Yuchen, Elishakoff, Isaac, Challamel, Noël, Florida Atlantic University, Department of Ocean and Mechanical Engineering, College of Engineering and Computer Science
Abstract/Description
In this thesis, we will explore different kinds of metamaterial or architectural structural problems, including structures composed of heterogeneous media with bi periodic sub-structures, discrete structures with sub-elements or continuous structures with discrete attached sub-elements. The thesis is composed of seven parts. After having introduced the specificities of metamaterial mechanics, the second chapter is devoted to the vibration of discrete beam problems called Hencky bar-chain...
Show moreIn this thesis, we will explore different kinds of metamaterial or architectural structural problems, including structures composed of heterogeneous media with bi periodic sub-structures, discrete structures with sub-elements or continuous structures with discrete attached sub-elements. The thesis is composed of seven parts. After having introduced the specificities of metamaterial mechanics, the second chapter is devoted to the vibration of discrete beam problems called Hencky bar-chain model in a stochastic framework. It is shown that the lattice beam behaves as a nonlocal continuous beam problem, both in the deterministic and the non-deterministic analyses. The third chapter considers the vibration of continuous beams with the introduction of shear effects and attached periodically oscillators. A discussion on beam modelling, for example Timoshenko beam models or truncated Timoshenko beam models is included. It is shown that the bandgap phenomenon observed for metamaterial beams can be accurately captured by a truncated Timoshenko beam model which means the last term in the Timoshenko equation is not that important.
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Date Issued
2023
PURL
http://purl.flvc.org/fau/fd/FA00014289
Subject Headings
Metamaterials, Hencky bar-chain model, Lattice theory, Engineering
Format
Document (PDF)